52 research outputs found
Simplicity of eigenvalues in Anderson-type models
We show almost sure simplicity of eigenvalues for several models of
Anderson-type random Schr\"odinger operators, extending methods introduced by
Simon for the discrete Anderson model. These methods work throughout the
spectrum and are not restricted to the localization regime. We establish
general criteria for the simplicity of eigenvalues which can be interpreted as
separately excluding the absence of local and global symmetries, respectively.
The criteria are applied to Anderson models with matrix-valued potential as
well as with single-site potentials supported on a finite box.Comment: 20 page
Smilansky's model of irreversible quantum graphs, I: the absolutely continuous spectrum
In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case is investigated. For the sake of
simplicity we consider K=2, but our argument is of a general character. In this
first of two papers on the problem, we describe the absolutely continuous
spectrum. Our approach is based upon scattering theory
Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
We give a self-contained presentation of the theory of self-adjoint
extensions using the technique of boundary triples. A description of the
spectra of self-adjoint extensions in terms of the corresponding Krein maps
(Weyl functions) is given. Applications include quantum graphs, point
interactions, hybrid spaces, singular perturbations.Comment: 81 pages, new references added, subsection 1.3 extended, typos
correcte
On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
We consider one-dimensional Schroedinger-type operators in a bounded interval
with non-self-adjoint Robin-type boundary conditions. It is well known that
such operators are generically conjugate to normal operators via a similarity
transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians
in quantum mechanics, we study properties of the transformations in detail. We
show that they can be expressed as the sum of the identity and an integral
Hilbert-Schmidt operator. In the case of parity and time reversal boundary
conditions, we establish closed integral-type formulae for the similarity
transformations, derive the similar self-adjoint operator and also find the
associated "charge conjugation" operator, which plays the role of fundamental
symmetry in a Krein-space reformulation of the problem.Comment: 27 page
9.4. The similarity problem and the structure of the singular spectrum of a nondissipative operator
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